Nonlinear Control and Planning in Robotics Lecture 10 : Lyapunov Redesign and Robust Backstepping April 6 , 2015

1 Uncertainty and Lyapunov Redesign Consider the system [1] ẋ = f(t, x) +G(t, x)[u+ δ(t, x, u)], (1) where x ∈ Rn is the state and u ∈ Rp is the control input. The functions f,G, and δ are defined for (x, u) ∈ D × Rp, where D ⊂ Rn contains the origin. The functions f,G and δ are piece-wise continuous and Lipschitz in x and u. We assume that f and G are known while δ is unknown and represents the combined effect of model simplification, parametric uncertainty, etc... [1]. When the uncertainty acts only along control vector fields (the columns of the matrix G) it is said to satisfy the matching condition, i.e. it matches the controls. The system (1) is in such form. Stabilizing controls can be designed for this case through the concept of Lyapunov redesign. In the nonmatching case, it is necessary to assume more restrictive assumptions about the bounds of δ and employ recursive techniques such as robust backstepping. A nominal model of the system is given by ẋ = f(t, x) +G(t, x)u, (2) and we assume that a feedback controller u = ψ(t, x) was designed so that the nominal closed-loop system ẋ = f(t, x) +G(t, x)ψ(t, x), (3) is uniformly asymptotically stable. Assume that the nominal control corresponds to a Lyapunov function V (t, x) such that α1(‖x‖) ≤ V (t, x) ≤ α2(‖x‖) (4) ∂tV + ∂xV · [f(t, x) +G(t, x)ψ(t, x)] ≤ −α3(‖x‖), (5) for all x ∈ D and where the functions αi are strictly increasing and satisfy αi(0) = 0 (such functions are said be class K functions). Assume that for u = ψ(x, t) + v, the uncertainty satisfies the bound ‖δ(t, x, ψ(t, x) + v)‖ ≤ ρ(t, x) + k0‖v‖, 0 ≤ k0 < 1 (6)